Considering that those extinction coefficients (whether provided by the manufacturer or calculated by some software) are approximations anyway, I think treating the stem part as a double helix and the loop part as a single strand and summing up the respective extinction coefficients gives a reasonably accurate result. A number of software for such calculations, based on the so-called nearest-neighbors method, is available online.

You can try measuring the absorbance of the hairpin at elevated temperature above its melting temperature.

I also suggest melting out the structure and measuring absorbance as signle-stranded nucleic acid

No matter what kind of structure, we always determine the absorbance in deionized water at elevated temperature (80°C) and calculate concentration summing up the standard extinction coefficients of each base.

Using the nearest neighbor method gives you the approximate extinction coefficient for single stranded oligomers at near neutral pH. Most software uses the nearest neighbor approximation. If your DNA forms a hairpin and you want to estimate its concentration by abosrbance you should heat it to > 80 °C to ensure that all secondary structures are melted out. This elimates many of the pi-stacking interactions that will alter the optical properties of the bases.

Also DNA is very flexible and can form multiple conformations and intra- and intermolecular interactions. If your DNA can form a hairpin it also can adopt a stable intermolecular duplex. To ensure that all of your DNA is adopting its lowest energy haripin structure it should be heated it to >80° and slowly cooled to the assay temperature to eliminate any kinetically trapped structural intermediates that will occur in any aqueous solution.

If your purpose is to calculate the concentration of DNA, then you don’t have to worry what kind of structure it adopts. Because, DNA concentration never changes based on the structure (of course, the extinction coefficient varies based on the structure it adopts). So, to calculate the DNA concentration based on the optical property, as others suggested you need to melt it, measure the absorbance and use the extinction coefficient derived from the nearest neighbour method to calculate the concentration of DNA. I think the following paper will guide you how to calculate the extinction coefficient:

Puglisi, J. D., and Tinoco, I. Jr. (1989) Absorbance melting curves of RNA. Methods Enzymol. 180, 304-325.

There may be many softwares available and even you can calculate it online. While, we have prepared an excel file to calculate the extinction coefficient based on nearest neighbour method. I may not wish to share that file, however, I can calculate extinction coefficient if you send me your sequences, in case if it is necessary.

and finally, all the best!!!